On the distribution of runners on a circle

Abstract

Consider n runners running on a circular track of unit length with constant speeds such that k of the speeds are distinct. We show that, at some time, there will exist a sector S which contains at least |S|n+Ω(k) runners. The bound is asymptotically tight up to a logarithmic factor. The result can be generalized as follows. Let f(x,y) be a complex bivariate polynomial whose Newton polytope has k vertices. Then there exist a∈ℂ∖{0} and a complex sector S={reıθ:r>0,α≤θ≤β} such that the univariate polynomial f(x,a) contains at least β−α2πn+Ω(k) non-zero roots in S (where n is the total number of such roots and 0≤(β−α)≤2π). This shows that the Real τ-Conjecture of Koiran (2011) implies the conjecture on Newton polytopes of Koiran et al. (2015).